Semiclassical theory for quantum quenches in finite transverse Ising chains

We present a quantitative semiclassical theory for the nonequilibrium dynamics of transverse Ising chains after quantum quenches, in particular, sudden changes of the transverse field strength. We obtain accurate predictions for the quench-dependent relaxation times and correlation lengths, and also about the recurrence times and quasiperiodicity of time-dependent correlations in finite systems with open or periodic boundary conditions. We compare the quantitative predictions of our semiclassical theory (local magnetization, equal-time bulk-bulk and surface-to-bulk correlations, and bulk autocorrelations) with the results from exact free-fermion calculations, and discuss the range of applicability of the semiclassical theory and possible generalizations and extensions.

Figure 1

Typical semiclassical contribution to the correlation function $C(r_1,t_1;r_2,t_2)$. Note that the six trajectories of the three QP pairs intersect the line $(r_1,t_1;r_2,t_2)$ five, i.e., an odd number, of times, which implies that ${\sigma }_{r_1}^z\left(t_1\right)$ and ${\sigma }_{r_2}^z\left(t_2\right)$ have opposite orientation. Equivalently, one can say that the trajectories of the red and the green QP pair intersect $(r_1,t_1;r_2,t_2)$ an even number of times (and thus do not contribute) and the trajectory of the blue QP pair an odd number of times.

Figure 2

Left: Typical semiclassical contribution to the time dependence of the local magnetization $m_l\left(t\right)$. Full lines are quasiparticles or kinks moving with velocity $v_p$ through the chain. The $\pm$ signs denote the sign of the spin at site $l$. Right: Sketch of the trajectories of kink pairs that flip the spin at position $l$ exactly once for times $t<T_p/2$. Kink pairs with initial position $x_0$ outside the marked region either do not flip the spin at $l$ (since they do not reach the position $l$ within time $t$) or they flip it twice. $q_p$ is the fraction of the marked intervals on the $t=0$ axis.

Figure 3

Semiclassical prediction for the local magnetization $m_l\left(t\right)$ quench. Here, $L=1024$, $h_0=0$, $h=0.2$. The (quasi)periodicity (20) is $T_{\mathrm{period}}=L/h=5120$.

Figure 4

Relaxation of the local magnetization $\log m_l\left(t\right)$ at different positions in a $L=256$ chain with free ends after a quench with parameters $h_0=0.0$, $h=0.2$, and $L=256$. (a) Exact (free-fermion calculation). (b) Semiclassical prediction (15) with the passing probability (18) and the occupation probability (A8). (c) Comparison between exact and QP calculation for $m_l\left(t\right)$ for $L=256$, $l=128$ for a quench from $h_0=0$ to $0.1$. (d) Semiclassical prediction using a thermal occupation number probability in Eq. (21) with an effective temperature $T_{\mathrm{eff}}$ (see the text).

Figure 5

Relaxation of the local magnetization $m_l\left(t\right)$ after quenches into the disordered phase, from the ordered phase ($h_0=0.5$, $h=1.5$) (a) exact, (b) semiclassical, and from the disordered phase ($h_0=1.5$, $h=2.0$) (c) exact, (d) semiclassical. The legend of (a) and (b) holds also for (c) and (d), the system size is $L=256$.

Figure 6

Semiclassical contributions to the equal-time correlation function $C_t\left(r\right)=C(L/2-r/2,t;L/2+r/2,t)$. Left: Sketch of the trajectories of kink pairs that reverse the orientation of the spins at $r_1$ and $r_2$ for times $t<T_p/4$. Kink pairs with initial position $x_0$ outside the marked region either do not intersect the line $(r_1=L/2-r/2,t;r_2=L/2+r/2,t)$ (red) (since they do not reach the red line within the time $t$) or they flip it twice. $q_p^c$ is the fraction of the marked intervals on the $t=0$ axis. Right: Sketch of the additional symmetry of $q_p^c\left(t\right)$ that reduce its periodicity from $T_p$ to $T_p/2$. For each QP pair created at position $x_0$ intersection, the red line at time $t<T_p/4$ there is a QP pair created at position $L-x_0$ that intersects the red line at time $T_p/2-t$. Hence, $q_p^c\left(t\right)=q_p^c(T_p/2-t)$ for $t<T_p/2$. At $t=T_p/2$, the QP pair created at $x_0$ meets again at $L-x_0$ and the one created at $L-x_0$ meets again at $x_0$, which implies after averaging over initial position that $q_p^c(t+T_p/2)=q_p^c\left(t\right)$.

Figure 7

Equal-time correlation function $C_t\left(r\right)$ for fixed $r$ as a function of time $t$ after the quench: comparison between the exact result (left) and the semiclassical prediction (right). $L=256$, $h_0=0$, $h=1.5$ (a) exact, (b) semiclassical; $h_0=0.3$, $h=0.5$ (c) exact, (d) semiclassical. The legend of (a) and (b) holds also for (c) and (d).

Figure 8

Equal-time correlation function $C_t\left(r\right)$ for fixed time $t$ after the quench as a function of distance $r$: comparison between the exact result (points) and the QP calculation (black broken lines). $L=256$, $h_0=0$, and $h=0.25$.

Figure 9

QP prediction for the equal-time correlation function $C_t\left(r\right)$ plotted against time after the quench scaled with the (quasi)period $T_{\mathrm{period}}^C=L/2h$ for different fields $h$ ($h_0=0$). The three curves for each field $h$ correspond to different system sizes: $L=256$, 512, and 1024 (from the top curve to the bottom curve).

Figure 10

Surface-to-bulk correlation function $C_t^{\mathrm{surf}}\left(r\right)$ for fixed time $r$ as a function of the time after the quench. Comparison between the exact result (points) and the QP calculation (black broken lines). $L=256$, $h_0=0.75$, and $h=0.25$.